OFCE
mercredi 13 novembre 2024
Overview of the model
Main equations
Multi-sector Macroeconomic Model for the Evaluation of Environmental and Energy policy
Macroeconomic multi-sector model with neo-keynesian features
Open source model: www.threeme.org
n sectors (e.g. France: n = 37; Tunisia: n = 21, Mexico: n = 29, Luxembourg: n = 26)
Allows to analyze the effect of transfer of activities from one sector to another on:
The economy is disaggregated into 27 sectors, with in particular:
and 23 commodities
The disaggregation is a compromise between the availability of the national account and energy data and the objective of the model
Several sectors have to be disaggregated
The energy disaggregation allows for analyzing the energy behavior of economic agents:
Energy transition policies
External shock
Computable: numerical simulation
General: take into account the interactions between markets.
Equilibrium: Supply is equal to demand on all markets (good, production factors)
Structure of a CGE model (see next Figure):
General Equilibrium relates to a state where supply is equal to demand in all markets
2 main approaches to ensure this state:
The equilibrium force is the price system
Perfect price flexibility ensures the instantaneous equilibrium between supply and demand
When the supply of a commodity goes down, its price tends to go up, thereby stimulating additional supply and depressing demand, until supply and demand are equal again.
Static model
Slow adjustment of prices and quantities
Leads to situation of disequilibrium between the desired supply and demand
Prices are defined as a mark-up over the firm’s production costs
Wages are determined by a Wage Setting (WS) curve
The interest rate does not equilibrate instantaneously saving and investment:
General equilibrium effects
Direct and indirect effects of the energy transition
Limited eviction effects
Sectorial disaggregation
High technological disaggregation of the energy system
General Equilibrium
Neo-keynesian features
Hybrid modelling
Possibility of combination with an energy system model
SU table says how much a given commodities is supplied by a given sector (Supply)
IO table says how much a given commodities is purchased by a given sector (Demand)
Overview of the model
Main equations
ThreeME takes into account explicitly the slow adjustment of prices and quantities (factors of production, consumption)
ThreeME assumes that the actual levels of prices and quantities gradually adjust to their notional level
The notional level corresponds to the optimal (desired or target) level that the economic agent would choose in the absence of adjustment constraints
Microeconomic foundation for a slow adjustment process:
Adjustment models: Eisner and Strotz (1963) and Gould (1968), Lucas (1967), Schramm (1970) and Treadway (1971), Rotemberg (1982)
The adjustment process is derived from an optimal behavior: minimizing an adjustment cost function (Rotemberg,1982)
\[\operatorname{log} X_{t} = \alpha^{{0},X} \; \operatorname{log} X^{n}_{t} + \left( 1 - \alpha^{{0},X} \right) \; \left( \operatorname{log} X_{t-1} + \varDelta \left(\operatorname{log} X^{e}_{t}\right) \right)\]
Expectations for prices and quantities
\[\varDelta \left(\operatorname{log} X^{e}_{t}\right) = \alpha^{{1},X} \; \varDelta \left(\operatorname{log} X^{e}_{t-1}\right) + \alpha^{{2},X}\; \varDelta \left(\operatorname{log} X_{t-1}\right) \\ + \alpha^{{3},X} \; \varDelta \left(\operatorname{log} X^{n}_t\right) + \alpha^{{4},X} \; \varDelta \left(\operatorname{log} X_{t+1}\right)\]
Where \(X_{t}\) is the effective value of a given variable, \(X^{n}_t\) is its notional level, \(X^{e}_t\) its expected (anticipated) value at period \(t\) and \(\alpha^{{i},X}\) are the adjustments parameters, with \(\alpha^{{1},X} + \alpha^{{2},X} + \alpha^{{3},X} + \alpha^{{4},X} = 1\).
In the long run, ThreeME converges toward a steady state where all variables grow at a constant rate:
ThreeME is not a so-called ”new Keynesian” model as defined in the Dynamic Stochastic General Equilibrium (DSGE) literature. In particular, ThreeME does not assume a functioning of the economy with perfect information (about the future) and with perfectly rational and forward-looking agents.
\[\begin{equation} \Gamma_t\left(X_t\right)-\Gamma_t\left(X_t^n\right)=\Gamma_t^{\prime}\left(X_t^n\right)\left(X_t-X_t^n\right)+\Gamma_t^{\prime \prime}\left(X_t^n\right)\left(X_t-X_t^n\right)^2 \end{equation}\] The profit being maximum for \(X_t^n, \Gamma_t^{\prime}\left(X_t^n\right)=0\) and \(\Gamma_t^{\prime \prime}\left(X_t^n\right)<0\). As a first approximation, the adjustment cost, i.e. the loss of profit suffered by a company that is not in the optimum, is therefore: \[ C_D=\Gamma_t\left(X_t^n\right)-\Gamma_t\left(X_t\right)=C_D\left(X_t-X_t^n\right)^2 \] Where: \[ C_D=-\Gamma_t^{\prime \prime}\left(X_t^n\right) \] Suppose that the adjustment cost is proportional to the square of the speed of adjustment: \[ C_A=c_A\left(X_t-X_{t-1}\right)^2 \] Where: \[ c_A>0 \] Minimizing the total cost function \(\left(C_t=C_D+C_A\right)\) is equivalent to solving: \[ C_t^{\prime}\left(X_t\right)=2 c_D\left(X_t-X_t^n\right)+2 c_A\left(X_t-X_{t-1}\right)=0 \] The condition of the second order \(\left(C_t^{\prime \prime}\left(X_t\right)>0\right)\) being always verified, the optimal adjustment which minimizes the total cost has the following dynamic process: \[ X_t=\alpha X_t^n+(1-\alpha) X_{t-1} \] With: \[ \alpha=\frac{c_D}{\left(c_D+c_A\right)} \]
With this simple model, the average adjustment time is: \[ \frac{\alpha}{(1-\alpha)}=\frac{c_D}{c_A} \] The slower the adjustment, the higher the adjustment cost \(c_A\) compared to the cost of non being adjusted \(c_D\).
\[Y_{c, s} = \varphi^{Y}_{c, s} \; YQ_{c}\]
\(YQ_c\) : aggregated domestic production of commodity \(c\). It is determined by the demand (intermediate and final consumption, investment, public spending, exports and stock variation).
\(PhiY_{c, s}\) : share of commodity \(c\) produced by the sector \(s\) (with \(\sum_{s} phiY_{c,s} =1\))
Aggregated production of sector \(s\):
\[Y_{s} = \sum_{s} Y_{c, s}\]
Notional production factors demand derived from a production costs minimization program assuming a Variable Output Elasticity (VOE) Cobb–Douglas production function
\[\varDelta \left(\operatorname{log} F^{n}_{f, s}\right) = \varDelta \left(\operatorname{log} Y_{s}\right) - \varDelta \left(\operatorname{log} PROG_{f, s}\right) + \varDelta \left(SUBST^{F}_{f, s}\right)\]
\[\varDelta \left(SUBST^{F,n}_{f, s}\right) = \sum_{ff} -ES_{f, ff, s} \; \varphi_{ff, s, t-1} \; \varDelta \left(\operatorname{log} \frac{C_{f, s}}{PROG_{f, s}} - \operatorname{log} \frac{C_{ff, s}}{PROG_{ff, s}}\right)\]
\[ES_{K, E, s} = \frac{\eta^{NEST^{K,E}}_{s}}{\left( 1 - \varphi_{MAT, s} - \varphi_{L, s} \right) - \frac{\eta^{NEST^{MAT,KEL}}_{s}}{\left( 1 - \varphi_{MAT, s} \right)} - \frac{\eta^{NEST^{KE,L}}_{s} \;\varphi_{L, s}}{{(1 - \varphi_{MAT, s})}{(1 - \varphi_{MAT, s} - \varphi_{L, s})}}}\]
\[ES_{K, L, s} = \frac{\eta^{NEST^{KE,L}}_{s} - \eta^{NEST^{MAT,KEL}}_{s} \; \varphi_{MAT, s}}{1 - \varphi_{MAT, s}}\]
Investment depends:
\[\varDelta \left(\operatorname{log} IA_{s}\right) = \alpha^{IA,Ye}_{s} \; \varDelta \left(\operatorname{log} Y^{e}_{s}\right) + \alpha^{IA,IA1}_{s} \; \varDelta \left(\operatorname{log} IA_{s, t-1}\right) \\ + \alpha^{IA,SUBST}_{s} \; \varDelta \left(SUBST^{F}_{K, s}\right) \\ + \alpha^{IA,Kn}_{s} \; \left( \operatorname{log} F^{n}_{K, s, t-1} - \operatorname{log} F_{K, s, t-1} \right)\]
\[F_{K, s} = \left( 1 - \delta_{s} \right) \; F_{K, s, t-1} + IA_{s}\]
The notional production price for each sector is set by applying a mark-up over the unit cost of production
\[PY^{n}_{s} = CU^{n}_{s} \; \left( 1 + \mu_{s} \right)\]
Where \(PY^n\) is the notional price, \(CU^n_s\) the unitary cost of production and \(\mu_{s}\) is the mark-up.
\[\varDelta \left(\operatorname{log} \left(1 + \mu^{n}_{s}\right)\right) = \rho^{\mu,Y} . \varDelta \left(\operatorname{log} CUR_{s}\right)\]
capacity utilization ratio \[CUR_{s} = \frac{Y_{s}}{YCAP_{s}}\]
Production capacity
\[\varDelta \left(\operatorname{log} YCAP_{s}\right) = \sum_{f} \varphi_{f, s, t-1} \; \varDelta \left(\operatorname{log} \left(F_{f, s} \; PROG_{f, s}\right)\right) \]
\[\varDelta \left(\operatorname{log} W^{n}_{s}\right) = \rho^{W,Cons}_{s} + \rho^{W,P}_{s} \; \varDelta \left(\operatorname{log} P\right) + \rho^{W,Pe}_{s} \; \varDelta \left(\operatorname{log} P^{e}\right) \\ + \rho^{W,PROG}_{s} \; \varDelta \left(\operatorname{log} PROG^{L}_{s}\right) - \rho^{W,U}_{s} \; \left( UnR - DNAIRU \right) \\ - \rho^{W,DU}_{s} \; \varDelta \left(UnR\right) + \rho^{W,L}_{s} \; \varDelta \left(\operatorname{log} F_{L, s} - \operatorname{log} F_{L}\right)\]
General specification that includes the Phillips and Wage Setting (WS) curves depending on the value of the selected parameters (Heyer et al., 2007; Reynès, 2010)
In the neo-Keynesian framework, it is standard to assume that the interest rate is set by the Central Bank (CB) according to a Taylor rule (Taylor, 1993).
The CB maximizes an objective function by making a trade-off between inflation and economic activity (unemployment)
The real interest rate is
\[\varDelta \left(R^{n}\right) = \rho^{Rn,P} . \varDelta \left(\frac{\varDelta \left(P\right)}{P_{t-1}}\right) - \rho^{Rn,UnR} . \varDelta \left(UnR\right)\]
\[CH^{n,VAL} = DISPINC^{AT,VAL} . \left( 1 - MPS^{n} \right)\]
Where \(CH^{n,VAL}\) is the aggregate notional households final consumption expressed in value, \(DISPINC^{AT,VAL}\) is the households’ disposable income - The notional marginal propensity to save (\(MPS^n\) their ) depends on the real interest rate and on the unemployment rate:
\[\varDelta \left(MPS^{n}\right) = \rho^{MPS,R} . \varDelta \left(R - \frac{\varDelta \left(P\right)}{P_{t-1}}\right) + \rho^{MPS,UnR} . \varDelta \left(UnR\right)\]
ThreeME includes three model variants for the allocation of the aggregate consumption between the different commodities:
In the standard version of the model, consumption decisions between commodities are modeled through a Linear Expenditure System (LES) utility function generalized to the case of a non-unitary elasticity of substitution between the commodities (Brown and Heien, 1972).
A LES specification assumes that a share of the base year consumption (\(NCH_c\)) is incompressible: the relation between income and consumption is not linear
This specification allows for the distinction between the consumption of necessity and luxurious goods
Notional consumption of commodity \(c\): \[\left( CH^{n}_{c} - NCH_{c} \right) \; PCH_{c} = \varphi^{MCH}_{c} \; \left( CH^{n,VAL} - PNCH . NCH \right)\]
Share of consumption of commodity \(c\):
\[\varDelta \left(\operatorname{log} \varphi^{MCH}_{c}\right) = \left( 1 - \eta^{LESCES} \right) . \varDelta \left(\operatorname{log} \frac{PCH_{c}}{PCH^{CES}}\right)\]
\[PCH^{CES} = \left( \sum_{c} \varphi^{MCH}_{c, t_0} \; PCH_{c} ^ {\left( 1 - \eta^{LESCES} \right)} \right) ^ {\left( \frac{1}{\left( 1 - \eta^{LESCES} \right)} \right)}\]
\[\varDelta \left(\operatorname{log} X_{c}\right) = \varDelta \left(\operatorname{log} WD_{c}\right) + \varDelta \left(SUBST^{X}_{c}\right)\]
\[\varDelta \left(SUBST^{X,n}_{c}\right) = -\eta^{X}_{c} \; \varDelta \left(\operatorname{log} PX_{c} - \operatorname{log} \left(EXR . PWD_{c}\right)\right)\]
Where \(WD_{c}\) is the world demand, \(PWD_{c}\) its price. \(PX_{c}\) is the export price that depends on the production costs and which reflects the price-competitiveness of the domestic products. \(EXR\) is the exchange rate; \(\sigma^{X}_{c}\) is the price-elasticity (assumed constant).
\[AM_{c} = \varphi^{AM}_{c} \; A_{c}\]
\[AD_{c} = \left( 1 - \varphi^{AM}_{c} \right) \; A_{c}\]
\[\varphi^{AM}_{c} = \frac{1}{\left( 1 + \frac{AD_{c}}{AM_{c}, t_0} \; \operatorname{exp} SUBST^{AM}_{c} \right)}\] \[\varDelta \left(SUBST^{AM,n}_{c}\right) = -\sigma^{AM}_{c} \; \varDelta \left(\operatorname{log} PAD_{c} - \operatorname{log} PAM_{c}\right)\]
\[SUBST^{AM}_{c} = \alpha^{{6},AM}_{c} \; SUBST^{AM,n}_{c} + \left( 1 - \alpha^{{6},AM}_{c} \right) \; SUBST^{AM}_{c, t-1}\]
\[INC^{G,VAL} = PNTAXC . NTAXC + NTAXS^{VAL} + INC^{SOC,TAX,VAL} \\ + PRSC . RSC + PROP^{INC,G,VAL}\]
with:
\[SPEND^{G,VAL} = PG . G + SOC^{BENF,VAL} + DEBT^{G,VAL}_{t-1} \; \left( \varphi^{RD^{G}}_{t-1} + r^{DEBT,G}_{t-1} \right)\]
with:
Emissions of the greenhouse gas \(ghg\) related to the intermediary consumption of commodity \(c\) by sector \(s\)
\[\varDelta \left(\operatorname{log} EMS^{CI}_{ghg, c, s}\right) = \varDelta \left(\operatorname{log} \left(CI_{c, s} \; IEMS^{CI}_{ghg, c, s}\right)\right)\]
where \(IEMS^{CI}_{ghg, c, s}\) is the corresponding emission intensity calibrated to 1 in the base year. It may change over time because of the increase of the share of biofuels.
Emissions of the greenhouse gas \(ghg\) related to the household consumption \(c\)
\[\varDelta \left(\operatorname{log} EMS^{CH}_{ghg, c}\right) = \varDelta \left(\operatorname{log} \left(CH_{c} \; IEMS^{CH}_{ghg, c}\right)\right)\]
Emissions of the greenhouse gas \(ghg\) related to the final production of sector \(s\)
\[\varDelta \left(\operatorname{log} EMS^{Y}_{ghg, s}\right) = \varDelta \left(\operatorname{log} \left(Y_{s} \; IEMS^{Y}_{ghg, s}\right)\right)\]
Emissions of the greenhouse gas \(ghg\) related to the materials consumption of sector \(s\)
\[\begin{equation} \small \varDelta \left(\operatorname{log} EMS^{MAT}_{ghg, s}\right) = \varDelta \left(\operatorname{log} \left(F_{MAT, s} \; IEMS^{MAT}_{ghg, s}\right)\right) \end{equation}\]
\(CO_2\) emissions from decarbonation during the production process for the non mineral metallic products, as the glass or ceramic sectors, is assumed proportional to the quantity of intermediate raw material used in the production process.
\[\varDelta \left(\operatorname{log} CI^{toe}_{ce, s}\right) = \varDelta \left(\operatorname{log} CI_{ce, s}\right)\]
\[Y^{toe}_{ce} + M^{toe}_{ce} = CI^{toe}_{ce} + CH^{toe}_{ce} + X^{toe}_{ce}\]
\[Y^{toe}_{ce, s} = \varphi^{Y^{toe}}_{ce, s} \; Y^{toe}_{ce}\] where \(\varphi_{ce,s}^{Y^{toe}}\) is the share energy \(ce\) produced by sector \(s\)
\[\varphi^{Y}_{ce, s} = \frac{PY^{toe}_{ce, s, t_0} \; \varphi^{Y^{toe}}_{ce, s}}{\sum_{ss} \left( PY^{toe}_{ce, ss, t_0} \; \varphi^{Y^{toe}}_{ce, ss} \right)}\]
A presentation of ThreeME